Kirwan surjectivity for the equivariant Dolbeault cohomology

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Abstract

Consider the holomorphic Hamiltonian action of a compact Lie group K on a compact Kähler manifold M with a moment map Φ:M→k. Assume that 0 is a regular value of the moment map. Weitsman raised the question of what we can say about the cohomology of the Kähler quotient M0≔Φ−1(0)∕K if all the ordinary cohomology of M is of type (p,p). In this paper, using the Cartan–Chern–Weil theory we show that in the above context there is a natural surjective Kirwan map from an equivariant version of the Dolbeault cohomology of M onto the Dolbeault cohomology of the Kähler quotient M0. As an immediate consequence, this result provides an answer to the question posed by Weitsman.

Original languageEnglish
Pages (from-to)43-53
Number of pages11
JournalJournal of Geometry and Physics
Volume144
DOIs
StatePublished - Oct 2019

Scopus Subject Areas

  • Mathematical Physics
  • General Physics and Astronomy
  • Geometry and Topology

Keywords

  • Cartan–Chern-Weil theory
  • Kähler quotient
  • Moment maps

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